单形的第一Cheeger常数
The first Cheeger constant of a simplex
摘要：
共边界扩展将经典的图扩展推广到一般单纯复形的情况，并允许定义任意单纯复形$X$和任何$k\geq 0$的高维Cheeger常数$h_k(X)$。本文研究了具有$n$顶点的单形的第一Cheeger常数$h_1(Δ^{[n]})$的值。由于Meshulam和Wallach的开创性工作，已知当$n$可被$3$整除时，实现了\[\lceil n/3\rceil\geq h_1(Δ^{[n]})\geq n/3, \textrm{ for all } n,\]和等式$h_1(Δ^{[n]})=n/3$。在这里，我们扩展这些结果。首先，我们证明了\[h_1(Δ^{[n]})=n/3, \textrm{ whenever }n\textrm{ is not a power of }2.\]所以在密度为$1$的集合上，尖锐的等式成立。第二，我们证明了\[h_1(Δ^{[n]})=n/3+O(1/n),\textrm{ when }n\textrm{ is a power of }2.\]换句话说，当$n$趋于无穷大时，值$h_1(Δ^{[n]})-n/3$要么是$0$，要么非常迅速地变为$0$。我们的方法包括重铸原来的问题，在纯粹的图论语言，其次是一个特定的图形家庭，所谓的{ \it楼梯图}的详细调查。这些是通过将图与每个分区相关联来定义的，并且似乎特别适合于获得关于单形的第一个Cheeger常数的信息。
英文摘要：
The coboundary expansion generalizes the classical graph expansion to the case of the general simplicial complexes, and allows the definition of the higher-dimensional Cheeger constants $h_k(X)$ for an arbitrary simplicial complex $X$, and any $k\geq 0$. In this paper we investigate the value of $h_1(Δ^{[n]})$ - the first Cheeger constant of a simplex with $n$ vertices. It is known, due to the pioneering work of Meshulam and Wallach, that \[\lceil n/3\rceil\geq h_1(Δ^{[n]})\geq n/3, \textrm{ for all } n,\] and that the equality $h_1(Δ^{[n]})=n/3$ is achieved when $n$ is divisible by $3$. Here we expand on these results. First, we show that \[h_1(Δ^{[n]})=n/3, \textrm{ whenever }n\textrm{ is not a power of }2.\] So the sharp equality holds on a set whose density goes to $1$. Second, we show that \[h_1(Δ^{[n]})=n/3+O(1/n),\textrm{ when }n\textrm{ is a power of }2.\] In other words, as $n$ goes to infinity, the value $h_1(Δ^{[n]})-n/3$ is either $0$ or goes to $0$ very rapidly. Our methods include recasting the original question in purely graph-theoretic language, followed by a detailed investigation of a specific graph family, the so-called {\it staircase graphs}. These are defined by associating a graph to every partition, and appear to be especially suited to gain information about the first Cheeger constant of a simplex.
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